COMPARING INTERVALS
AND MOMENTS FOR THE
QUANTIFICATION OF
COARSE INFORMATION
M. Beer
University of Liverpool
V. Kreinovich
University of Texas at El Paso
Michael Beer, Vladik Kreinovich
1 / 12
1
Problem description
COARSE INFORMATION
 linguistic assessments
 measuring devices
2
high
4
medium
6
5.15 ... 5.35
low
0
10
30
50
D [N/mm²]
 expert assessment / experience
 measuring points
plausible range
thickness
Michael Beer, Vladik Kreinovich
d
x
measurement / observation
under dubious conditions
2 / 12
1
Problem description
CLASSIFICATION AND MODELING
According to sources
 aleatory uncertainty
 epistemic uncertainty
» irreducible uncertainty
» property of the system
» fluctuations / variability
» reducible uncertainty
» property of the analyst
» lack of knowledge or perception
stochastic characteristics
collection of all problematic cases,
inconsistency of information
no specific model
traditional
probabilistic models
According to information content
 uncertainty
» probabilistic information
traditional and subjective
probabilistic models
 imprecision
» non-probabilistic characteristics
set-theoretical models
In view of the purpose of the analysis
 averaged results, value ranges, worst case, etc. ?
Michael Beer, Vladik Kreinovich
3 / 12
3
Engineering comparison
PROBLEM CONTEXT
Structural reliability problem
 performance function
G  .  c  
N
c  6.35 cm
 P  p 
CcH
log  o

1  e0
P
o


 further example and detailed discussion
Beer, M., Y. Zhang, S. T. Quek, K. K. Phoon
Reliability analysis with scarce information:
Comparing alternative approaches in a
geotechnical engineering context
Structural Safety 41 (2013), 1–10.
» coarse information about the six variables Xi
Quantification of uncertain variables
 specification of 2 parameters
Type and amount of available information ? Purpose of analysis ?
» moments μ and σ2
probabilistic analysis, response moments, cdf, Pf
» interval bounds xil and xiu
interval analysis, range, worst case
Comparative study
 assume normal distribution for the variables
 relate interval bounds to moments: xil , xiu   X  3X ,  X  3X 
i
Michael Beer, Vladik Kreinovich
i
i
i
4 / 12
3
Engineering comparison
INTERPRETATION OF RESULTS
Probabilistic analysis
Interval analysis
 Pf  P G  .  0   8.94  104
 gl , gu   9.66 , 6.24  9.66 , 0  0, 6.24
Given that input information is coarse
failure may occur in a
moderate number of cases
failure may occur
comparable
magnitude of exceedance
of g = 0 rather small, strong
exceedance quite unlikely
significant exceedance of g = 0
may occur
different focus: consider low-probability-but-high-consequence events
General relationship
 bounding property P  Y  yl , yu   P  X  xl , xu  for general mapping XY
» known distribution of X
conclusions from
interval analysis mostly
too conservative
Michael Beer, Vladik Kreinovich
» unknown distribution of X
probabilistic results may be too
optimistic, worst case (which is
emphasized in interval analysis)
5 / 12
maybe likely
3
Engineering comparison
RELATIONSHIP BETWEEN RESULTS
Probabilistic analysis
Interval analysis
 normal distributions for all
 xil , xiu   X  3X ,  X  3X  for all Xi
variables Xi
histogram for G(.)
P  G .  gl , gu   0.99993
» estimation of intervals [glP,guP]
with P  G .  glP , guP   0.98391
from histogram
i
i
i
i
P  Xi  xil , xiu  ,i  1,.., 6   0.98391
gl , gu   9.66 , 6.24
P  G .  gl , gu   0.98391
large difference due to low
probability density for small g(.),
but critical for failure
▪ both-sided
glP , guP central  1.15,5.53
▪ left-sided w.r.t. lower bound gl
gl , guP left  9.66 ,5.39
10
interval result is conservative
differences controlled by distribution of G(.)
Michael Beer, Vladik Kreinovich
0
10 g(.)
moderate difference
due to high probability
density at upper bounds
6 / 12
3
Engineering comparison
RELATIONSHIP BETWEEN RESULTS
Probabilistic approximation
Interval analysis
 using estimated moments of G(.)
 gl , gu   9.66 , 6.24
G  3.795, 2G  0.822
P  G .  gl , gu   0.98391
» Chebyshev’s inequality with


P G .  glP , guP Cheby  0.98391
glP , guP Cheby  3.35 ,10.94
10
interval result shifted towards
failure domain, even more
conservative than Chebyshev
interval result reflects tendency
of the distribution of G(.)
to left-skewness
for right-skewed distribution of G(.),
Chebychev‘s inequality may lead
to the more conservative result
Michael Beer, Vladik Kreinovich
0
interval analysis
10 g(.)
Chebyshev
histogram for G(.)
for uniform Xi
10
10 g(.)
7 / 12
3
Engineering comparison
INTERVAL OR MOMENTS ?
General remarks
 interval analysis heads for the extreme events,
whilst a probabilistic analysis yields probabilities for events
 for a defined confidence level P  Xi  xil , xiu  , interval analysis is more
conservative and independent of distributions of the Xi
 conservatism of interval analysis is comparable to Chebyshev‘s inequality
 difference between interval results and probabilistic results
is controlled by the distribution of the response
 for a defined confidence level, interval bounds maybe easier to specify
or to control than moments
interval analysis can be helpful
» to identify low-probability-but-high-consequence events
for risk analysis
» in case of sensitivity of Pf w.r.t. distribution assumption and
very vague information for this assumption
» if the first 2 moments cannot be identified with sufficient confidence
What to chose in “intermediate” cases ?
Michael Beer, Vladik Kreinovich
8 / 12
4
Information-based comparison
INFORMATION CONTENT
Idea
 compare interval representation and moment representation
of uncertainty by means of information content:
Which representation tells us more ?
 assume that a variable X is represented alternatively
(i) by the first two moments μX and σX2
(ii) by an interval [xl, xu] for a given confidence P  X  xl , xu  
 apply maximum entropy principle to both representations;
calculate the least information of the representation
without making any additional assumptions
 chose the more informative representation;
exploit available information to maximum extent
(not in contradiction with maximum entropy principle)
Relating intervals and moments
 analog to the concept of confidence intervals
xl , xu   X  k  X , X  k  X 
Michael Beer, Vladik Kreinovich
9 / 12
4
Information-based comparison
ENTROPY-BASED COMPARISON
Shannon‘s entropy
 continuous entropy
S  f     f  x   log2  f  x   dx
» modification for comparison (ease of derivation)
log2  f  x   
ln  f  x  
Sm  f  
ln 2 
1
 S  f     f  x   ln  f  x   dx
ln 2 
Interval representation
 maximum entropy principle
uniform distribution
xu
Sm ,int   
xl
f x 
1
xu  xl
 1 
1
 ln 
 dx
xu  xl
x

x
l 
 u
 ln  xu  xl 
 relating to moments
xu  xl  2  k  X
Michael Beer, Vladik Kreinovich
Sm,int  ln 2  k  X   ln  X   ln 2  k 
10 / 12
4
Information-based comparison
ENTROPY-BASED COMPARISON
Moment representation
 maximum entropy principle
  x   2 
X

f x 
 exp  
normal distribution
2


2

2   X
X


1
Sm ,mom    f  x   ln  f  x   dx  ln  X   ln 2 
2
1


Comparison of representations
 check whether Sm,mom  Sm,int
ln  X   ln

ln


under the assumptions made
1
 ln  X   ln 2  k 
2
1
2   ln 2  k 
2
for k > 2,
the moment representation
is more informative
(ie, for >95% confidence)
2  e  2  k
for k ≤ 2,
the interval representation
is more informative
(ie, for <95% confidence)
2 

k 
Michael Beer, Vladik Kreinovich
e
 2.066
2
11 / 12
Comparing intervals and moments for the quantification of coarse information
CONCLUSIONS
Interval or moments
 depends on the problem and purpose of analysis
 for symmetric distributions, moment representation is
more informative if confidence of >95% is needed
 for skewed distributions, moment representation is already
more informative for smaller confidence
Remark 1: fuzzy sets
 nuanced consideration of a nested set of intervals
Remark 2: imprecise probabilities
 enable “intermediate” modeling between interval and cdf
 useful if probabilistic models are partly applicable
 consider a set of probabilistic models (eg interval parameters)
 worst case consideration in terms of probability (bounds)
Michael Beer, Vladik Kreinovich
12 / 12